Finite Group Databasebeta

← C5×#71688 #14283156

Subgroups — C5×#71688 #14283156

Order 1920.

588 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×C22⋊Q8C211yesmaximal, characteristic
2C4×C5×#217C211yesmaximal, characteristic
3C5×#5207C211yesmaximal, characteristic
4C5×#4538C214yesmaximal, complemented
5C5×#1508—31nomaximal, Hall
6C22×C3×C5×Q8C411yescharacteristic, complemented
7C23×C3×C4×C5C2211yescharacteristic
8C3×C5×C22⋊Q8—21no—
9C3×C5×C22⋊Q8—21no—
10C3×C5×C22⋊Q8C412yescomplemented
11C3×C5×C22⋊Q8C2212yes—
12C2×C3×C5×C4⋊C4—21no—
13C2×C3×C5×C4⋊C4C411yescharacteristic, complemented
14C2×C3×C5×C22⋊C4—21no—
15C2×C5×#217—21no—
16C2×C4×C5×Dic3—21no—
17C2×C5×#208—21no—
18C5×C3⋊#95C2212yes—
19C5×#830—21no—
20C5×#830C2212yes—
21#71688C511yesmaximal, characteristic, direct product factor, Hall
22C2×C5×C22⋊Q8D311yescharacteristic
23C5×#407—31no—
24C4×C5×C22⋊C4—31no—
25C5×#328—34no—
26C24×C3×C5D411yescharacteristic
27C2×C3×C5×Q8—21no—
28C2×C3×C5×Q8—42no—
29C2×C3×C5×Q8C2×C412yes—
30C22×C3×C4×C5—21no—
31C22×C3×C4×C5—21no—
32C22×C3×C4×C5—42no—
33C22×C3×C4×C5C2×C411yescharacteristic
34C22×C3×C4×C5C2×C411yescharacteristic
35C22×C3×C4×C5D411yescharacteristic
36C22×C3×C4×C5D411yescharacteristic
37C22×C3×C4×C5D411yescharacteristic
38C22×C3×C4×C5C2311yescharacteristic
39C22×C5×Dic3—22no—
40C3×C5×C4⋊C4—22no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4—22no—
43C3×C5×C4⋊C4C2×C412yes—
44C3×C5×C22⋊C4—22no—
45C3×C5×C22⋊C4—22no—
46C5×#217—24no—
47C4×C5×Dic3—42no—
48C5×#208—42no—
49C2×C5×C3⋊C8—22no—
50C2×C3×C22⋊Q8C2×C511yescharacteristic
51C4×#217C2×C511yescharacteristic
52#5207C2×C511yescharacteristic
53#4538C2×C514yescomplemented
54C22×C5×Q8Dic311yescharacteristic, complemented
55C23×C4×C5C2×D311yescharacteristic
56C2×C5×M16—61no—
57C5×C22⋊Q8—21no—
58C5×C22⋊Q8—21no—
59C5×C22⋊Q8Dic312yescomplemented
60C5×C22⋊Q8C2×D312yes—
61C2×C5×C4⋊C4—21no—
62C2×C5×C4⋊C4Dic311yescharacteristic, complemented
63C2×C5×C22⋊C4—21no—
64C2×C5×C22⋊C4—61no—
65C2×C42×C5—61no—
66C5×C22⋊C8—32no—
67C5×C22⋊C8—61no—
68C5×#95—32no—
69#1508—31noSylow
70C23×C3×C5—22no—
71C23×C3×C5—22no—
72C23×C3×C5—41no—
73C23×C3×C5C22⋊C411yescharacteristic
74C23×C3×C5C22⋊C411yescharacteristic
75C23×C3×C5C2×D411yescharacteristic
76C3×C5×Q8—42no—
77C3×C5×Q8—42no—
78C2×C3×C4×C5—22no—
79C2×C3×C4×C5—22no—
80C2×C3×C4×C5—22no—
81C2×C3×C4×C5—22no—
82C2×C3×C4×C5—22no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5—41no—
86C2×C3×C4×C5—42no—
87C2×C3×C4×C5—42no—
88C2×C3×C4×C5C22⋊C411yescharacteristic
89C2×C3×C4×C5C22⋊C411yescharacteristic
90C2×C3×C4×C5C22×C411yescharacteristic
91C2×C3×C4×C5C2×D411yescharacteristic
92C2×C5×Dic3—24no—
93C2×C5×Dic3—42no—
94C2×C5×Dic3—42no—
95C5×C3⋊C8—42no—
96C22×C3×Q8C4×C511yescharacteristic, complemented
97C23×C3×C4C22×C511yescharacteristic
98C3×C22⋊Q8—21no—
99C3×C22⋊Q8—21no—
100C3×C22⋊Q8C4×C512yescomplemented
101C3×C22⋊Q8C22×C512yes—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4C4×C511yescharacteristic, complemented
104C2×C3×C22⋊C4—21no—
105C2×#217—21no—
106C2×C4×Dic3—21no—
107C2×#208—21no—
108C3⋊#95C22×C512yes—
109#830—21no—
110#830C22×C512yes—
111C24×C5#6711yescharacteristic
112C2×C5×Q8—21no—
113C2×C5×Q8—42no—
114C2×C5×Q8C2×Dic312yes—
115C22×C4×C5—21no—
116C22×C4×C5—21no—
117C22×C4×C5—42no—
118C22×C4×C5—62no—
119C22×C4×C5C2×Dic311yescharacteristic
120C22×C4×C5C2×Dic311yescharacteristic
121C22×C4×C5#6711yescharacteristic
122C22×C4×C5#6711yescharacteristic
123C22×C4×C5#6711yescharacteristic
124C22×C4×C5C22×D311yescharacteristic
125C5×M16—122no—
126C2×C5×C8—62no—
127C5×C4⋊C4—22no—
128C5×C4⋊C4—21no—
129C5×C4⋊C4—22no—
130C5×C4⋊C4C2×Dic312yes—
131C5×C22⋊C4—22no—
132C5×C22⋊C4—22no—
133C5×C22⋊C4—64no—
134C42×C5—122no—
135C2×C22⋊Q8C5×D311yescharacteristic
136#407—31no—
137C4×C22⋊C4—31no—
138#328—34no—
139C22×C3×C5—21no—
140C22×C3×C5—21no—
141C22×C3×C5—21no—
142C22×C3×C5—21no—
143C22×C3×C5—21no—
144C22×C3×C5—21no—
145C22×C3×C5—21no—
146C22×C3×C5—44no—
147C22×C3×C5#10412yes—
148C22×C3×C5C2×C22⋊C411yescharacteristic
149C22×C3×C5#13712yes—
150C3×C4×C5—21no—
151C3×C4×C5—21no—
152C3×C4×C5—21no—
153C3×C4×C5—42no—
154C3×C4×C5—42no—
155C3×C4×C5#9912yes—
156C5×Dic3—44no—
157C24×C3C5×D411yescharacteristic
158C2×C3×Q8—21no—
159C2×C3×Q8—42no—
160C2×C3×Q8C2×C4×C512yes—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—42no—
164C22×C3×C4C2×C4×C511yescharacteristic
165C22×C3×C4C2×C4×C511yescharacteristic
166C22×C3×C4C5×D411yescharacteristic
167C22×C3×C4C5×D411yescharacteristic
168C22×C3×C4C5×D411yescharacteristic
169C22×C3×C4C23×C511yescharacteristic
170C22×Dic3—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4C2×C4×C512yes—
175C3×C22⋊C4—22no—
176C3×C22⋊C4—22no—
177#217—24no—
178C4×Dic3—42no—
179#208—42no—
180C2×C3⋊C8—22no—
181C23×C5—22no—
182C23×C5—22no—
183C23×C5—41no—
184C23×C5#21711yescharacteristic
185C23×C5#21711yescharacteristic
186C23×C5C2×#6711yescharacteristic
187C5×Q8—42no—
188C5×Q8—42no—
189C2×C4×C5—22no—
190C2×C4×C5—22no—
191C2×C4×C5—22no—
192C2×C4×C5—22no—
193C2×C4×C5—22no—
194C2×C4×C5—41no—
195C2×C4×C5—41no—
196C2×C4×C5—41no—
197C2×C4×C5—42no—
198C2×C4×C5—42no—
199C2×C4×C5—64no—
200C2×C4×C5—122no—
201C2×C4×C5—122no—
202C2×C4×C5#21711yescharacteristic
203C2×C4×C5#21711yescharacteristic
204C2×C4×C5C22×Dic311yescharacteristic
205C2×C4×C5C2×#6711yescharacteristic
206C5×C8—122no—
207C22×Q8C5×Dic311yescharacteristic, complemented
208C23×C4C2×C5×D311yescharacteristic
209C2×M16—61no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8C5×Dic312yescomplemented
213C22⋊Q8C2×C5×D312yes—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C5×Dic311yescharacteristic, complemented
216C2×C22⋊C4—21no—
217C2×C22⋊C4—61no—
218C2×C42—61no—
219C22⋊C8—32no—
220C22⋊C8—61no—
221#95—32no—
222C2×C3×C5—22no—
223C2×C3×C5—22no—
224C2×C3×C5—41no—
225C2×C3×C5C2×#9911yescharacteristic
226C2×C3×C5#41911yescharacteristic
227C2×C3×C5C2×#10411yescharacteristic
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—41no—
231C23×C3C5×C22⋊C411yescharacteristic
232C23×C3C5×C22⋊C411yescharacteristic
233C23×C3C2×C5×D411yescharacteristic
234C3×Q8—42no—
235C3×Q8—42no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4C5×C22⋊C411yescharacteristic
247C2×C3×C4C5×C22⋊C411yescharacteristic
248C2×C3×C4C22×C4×C511yescharacteristic
249C2×C3×C4C2×C5×D411yescharacteristic
250C2×Dic3—24no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C3⋊C8—42no—
254C22×C5—21no—
255C22×C5—21no—
256C22×C5—21no—
257C22×C5—21no—
258C22×C5—21no—
259C22×C5—21no—
260C22×C5—21no—
261C22×C5—44no—
262C22×C5#83712yes—
263C22×C5#94212yes—
264C22×C5C2×#21711yescharacteristic
265C4×C5—21no—
266C4×C5—21no—
267C4×C5—21no—
268C4×C5—42no—
269C4×C5—42no—
270C4×C5—124no—
271C4×C5#83412yes—
272C24C5×#6711yescharacteristic
273C2×Q8—21no—
274C2×Q8—42no—
275C2×Q8C2×C5×Dic312yes—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—62no—
280C22×C4C2×C5×Dic311yescharacteristic
281C22×C4C2×C5×Dic311yescharacteristic
282C22×C4C5×#6711yescharacteristic
283C22×C4C5×#6711yescharacteristic
284C22×C4C5×#6711yescharacteristic
285C22×C4C22×C5×D311yescharacteristic
286M16—122no—
287C2×C8—62no—
288C4⋊C4—22no—
289C4⋊C4—21no—
290C4⋊C4—22no—
291C4⋊C4C2×C5×Dic312yes—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—64no—
295C42—122no—
296C3×C5#150811yescharacteristic, complemented, Hall
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—44no—
305C22×C3C5×#10412yes—
306C22×C3C2×C5×C22⋊C411yescharacteristic
307C22×C3C5×#13712yes—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—42no—
312C3×C4—42no—
313C3×C4C5×#9912yes—
314Dic3—44no—
315C2×C5—22no—
316C2×C5—22no—
317C2×C5—41no—
318C2×C5C2×#83411yescharacteristic
319C2×C5#522511yescharacteristic
320C2×C5C2×#83711yescharacteristic
321C23—22no—
322C23—22no—
323C23—41no—
324C23C5×#21711yescharacteristic
325C23C5×#21711yescharacteristic
326C23C2×C5×#6711yescharacteristic
327Q8—42no—
328Q8—42no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—64no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4C5×#21711yescharacteristic
343C2×C4C5×#21711yescharacteristic
344C2×C4C22×C5×Dic311yescharacteristic
345C2×C4C2×C5×#6711yescharacteristic
346C8—122no—
347C2×C3—22no—
348C2×C3—22no—
349C2×C3—41no—
350C2×C3C2×C5×#9911yescharacteristic
351C2×C3C5×#41911yescharacteristic
352C2×C3C2×C5×#10411yescharacteristic
353C5#7168811yescharacteristic, direct product factor, Sylow
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—44no—
362C22C5×#83712yes—
363C22C5×#94212yes—
364C22C2×C5×#21711yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—42no—
369C4—42no—
370C4—124no—
371C4C5×#83412yes—
372C3C5×#150811yescharacteristic, complemented, Sylow
373C2—22no—
374C2—22no—
375C2—41no—
376C2C2×C5×#83411yescharacteristic
377C2C5×#522511yescharacteristic
378C2C2×C5×#83711yescharacteristic
subgroupG/H
centerC22×C5C2×#217
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
FittingC2×C3×C5×C22⋊Q8C2
socleC22×C3×C5C2×C22⋊C4
prime radicalC24×C3×C5D4
cosocleC22×C3×C4C23×C5
hypercenterC2×C5×C22⋊Q8D3
hypocenterC3C5×#1508
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C23×C4×C5 ◁ C2×C5×C22⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C5×#4538C2semidirect
6. C22×C3×C5×Q8C4semidirect
10. C3×C5×C22⋊Q8C4semidirect
13. C2×C3×C5×C4⋊C4C4semidirect
21. #71688C5direct (N × H)
53. #4538C2×C5semidirect
54. C22×C5×Q8Dic3semidirect
59. C5×C22⋊Q8Dic3semidirect
62. C2×C5×C4⋊C4Dic3semidirect
96. C22×C3×Q8C4×C5semidirect
100. C3×C22⋊Q8C4×C5semidirect
103. C2×C3×C4⋊C4C4×C5semidirect
207. C22×Q8C5×Dic3semidirect
212. C22⋊Q8C5×Dic3semidirect
215. C2×C4⋊C4C5×Dic3semidirect
296. C3×C5#1508semidirect
353. C5#71688direct (N × H)
372. C3C5×#1508semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×C5×C22⋊Q82
C4×C5×#2172
C5×#52072
C5×#45382
C5×#15083
#716885